Theorem cmptop 21198

Description: A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)

Assertion

Ref	Expression
cmptop	⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)

Proof of Theorem cmptop

Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscmp 21191	. 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑠)))
3	2	simplbi 476	1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∩ cin 3573  𝒫 cpw 4158  ∪ cuni 4436  Fincfn 7955  Topctop 20698  Compccmp 21189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-cmp 21190

This theorem is referenced by:  imacmp  21200  cmpcld  21205  fiuncmp  21207  cmpfii  21212  bwth  21213  locfincmp  21329  kgeni  21340  kgentopon  21341  kgencmp  21348  kgencmp2  21349  cmpkgen  21354  txcmplem1  21444  txcmp  21446  qtopcmp  21511  cmphaushmeo  21603  ptcmpfi  21616  fclscmpi  21833  alexsubALTlem1  21851  ptcmplem1  21856  ptcmpg  21861  evth  22758  evth2  22759  cmppcmp  29925  ordcmp  32446  poimirlem30  33439  heibor1lem  33608  cmpfiiin  37260  kelac1  37633  kelac2  37635  stoweidlem28  40245  stoweidlem50  40267  stoweidlem53  40270  stoweidlem57  40274  stoweidlem62  40279